Decay of shear layers and vortex sheets (Q1067507)

This work is an extension of a work previously done by the author [ibid. 14, 203-217 (1984; Zbl 0532.76003)] where time-dependent rectilinear flow of an incompressible viscoelastic fluid in a domain bounded by two stationary infinite rigid parallel plates was considered. In that paper, under the assumptions that (a) in an element of the fluid the shear stress depends linearly on the history of the velocity gradient (b) the fluid is at rest for \(t<0\) and at \(t=0\) it is subjected to a longitudinal pressure gradient which is subsequently held constant (c) the pressure gradient is a function of the distance from the mid-plane, an expression for the Laplace transform of the velocity field was obtained. With the aid of this expression, the author now proceeds in the present work to obtain the velocity field which results when the pressure gradient is such that the steady state velocity attained consists of two shear layers symmetrically disposed with respect to the mid-plane, in which the velocity gradients are equal in magnitude but opposite in sign, but constant in each shear layer. The results are then applied to the particular cases of Newtonian and Maxwellian fluids. There is quite a contrast between the results obtained. For Newtonian fluids, the velocity field spreads diffusively throughout the fluid while for the Maxwellian fluids, the velocity field may be interpreted as the superposition of eight waves which originate at \(t=0\) at the boundaries of the shear layers. Finally, two limiting cases are discussed. This is a well-written paper whose results should interest researchers in the field of viscoelastic fluids.